Simplify and expand the following expression: $ \dfrac{3}{4p - 8}+ \dfrac{5}{p + 3}+ \dfrac{3p}{p^2 + p - 6} $
Explanation: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $4$ out of denominator in the first term: $ \dfrac{3}{4p - 8} = \dfrac{3}{4(p - 2)}$ We can factor the quadratic in the third term: $ \dfrac{3p}{p^2 + p - 6} = \dfrac{3p}{(p - 2)(p + 3)}$ Now we have: $ \dfrac{3}{4(p - 2)}+ \dfrac{5}{p + 3}+ \dfrac{3p}{(p - 2)(p + 3)} $ The least common multiple of the denominators is: $ 4(p - 2)(p + 3)$ In order to get the first term over $4(p - 2)(p + 3)$ , multiply by $\dfrac{p + 3}{p + 3}$ $ \dfrac{3}{4(p - 2)} \times \dfrac{p + 3}{p + 3} = \dfrac{3(p + 3)}{4(p - 2)(p + 3)} $ In order to get the second term over $4(p - 2)(p + 3)$ , multiply by $\dfrac{4(p - 2)}{4(p - 2)}$ $ \dfrac{5}{p + 3} \times \dfrac{4(p - 2)}{4(p - 2)} = \dfrac{20(p - 2)}{4(p - 2)(p + 3)} $ In order to get the third term over $4(p - 2)(p + 3)$ , multiply by $\dfrac{4}{4}$ $ \dfrac{3p}{(p - 2)(p + 3)} \times \dfrac{4}{4} = \dfrac{12p}{4(p - 2)(p + 3)} $ Now we have: $ \dfrac{3(p + 3)}{4(p - 2)(p + 3)} + \dfrac{20(p - 2)}{4(p - 2)(p + 3)} + \dfrac{12p}{4(p - 2)(p + 3)} $ $ = \dfrac{ 3(p + 3) + 20(p - 2) + 12p} {4(p - 2)(p + 3)} $ Expand: $ = \dfrac{3p + 9 + 20p - 40 + 12p}{4p^2 + 4p - 24} $ $ = \dfrac{35p - 31}{4p^2 + 4p - 24}$